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101. CJM 2009 (vol 61 pp. 617)

Kim, Wook
Square Integrable Representations and the Standard Module Conjecture for General Spin Groups
In this paper we study square integrable representations and $L$-functions for quasisplit general spin groups over a $p$-adic field. In the first part, the holomorphy of $L$-functions in a half plane is proved by using a variant form of Casselman's square integrability criterion and the Langlands--Shahidi method. The remaining part focuses on the proof of the standard module conjecture. We generalize Mui\'c's idea via the Langlands--Shahidi method towards a proof of the conjecture. It is used in the work of M. Asgari and F. Shahidi on generic transfer for general spin groups.

Categories:11F70, 11F85

102. CJM 2009 (vol 61 pp. 674)

Pollack, David; Pollack, Robert
A Construction of Rigid Analytic Cohomology Classes for Congruence Subgroups of $\SL_3(\mathbb Z)$
We give a constructive proof, in the special case of ${\rm GL}_3$, of a theorem of Ash and Stevens which compares overconvergent cohomology to classical cohomology. Namely, we show that every ordinary classical Hecke-eigenclass can be lifted uniquely to a rigid analytic eigenclass. Our basic method builds on the ideas of M. Greenberg; we first form an arbitrary lift of the classical eigenclass to a distribution-valued cochain. Then, by appropriately iterating the $U_p$-operator, we produce a cocycle whose image in cohomology is the desired eigenclass. The constructive nature of this proof makes it possible to perform computer computations to approximate these interesting overconvergent eigenclasses.

Categories:11F75, 11F85

103. CJM 2009 (vol 61 pp. 373)

McKee, Mark
An Infinite Order Whittaker Function
In this paper we construct a flat smooth section of an induced space $I(s,\eta)$ of $SL_2(\mathbb{R})$ so that the attached Whittaker function is not of finite order. An asymptotic method of classical analysis is used.

Categories:11F70, 22E45, 41A60, 11M99, 30D15, 33C15

104. CJM 2009 (vol 61 pp. 395)

Moriyama, Tomonori
$L$-Functions for $\GSp(2)\times \GL(2)$: Archimedean Theory and Applications
Let $\Pi$ be a generic cuspidal automorphic representation of $\GSp(2)$ defined over a totally real algebraic number field $\gfk$ whose archimedean type is either a (limit of) large discrete series representation or a certain principal series representation. Through explicit computation of archimedean local zeta integrals, we prove the functional equation of tensor product $L$-functions $L(s,\Pi \times \sigma)$ for an arbitrary cuspidal automorphic representation $\sigma$ of $\GL(2)$. We also give an application to the spinor $L$-function of $\Pi$.

Categories:11F70, 11F41, 11F46

105. CJM 2009 (vol 61 pp. 336)

Garaev, M. Z.
The Large Sieve Inequality for the Exponential Sequence $\lambda^{[O(n^{15/14+o(1)})]}$ Modulo Primes
Let $\lambda$ be a fixed integer exceeding $1$ and $s_n$ any strictly increasing sequence of positive integers satisfying $s_n\le n^{15/14+o(1)}.$ In this paper we give a version of the large sieve inequality for the sequence $\lambda^{s_n}.$ In particular, we obtain nontrivial estimates of the associated trigonometric sums ``on average" and establish equidistribution properties of the numbers $\lambda^{s_n} , n\le p(\log p)^{2+\varepsilon}$, modulo $p$ for most primes $p.$

Keywords:Large sieve, exponential sums
Categories:11L07, 11N36

106. CJM 2009 (vol 61 pp. 264)

Bell, J. P.; Hare, K. G.
On $\BbZ$-Modules of Algebraic Integers
Let $q$ be an algebraic integer of degree $d \geq 2$. Consider the rank of the multiplicative subgroup of $\BbC^*$ generated by the conjugates of $q$. We say $q$ is of {\em full rank} if either the rank is $d-1$ and $q$ has norm $\pm 1$, or the rank is $d$. In this paper we study some properties of $\BbZ[q]$ where $q$ is an algebraic integer of full rank. The special cases of when $q$ is a Pisot number and when $q$ is a Pisot-cyclotomic number are also studied. There are four main results. \begin{compactenum}[\rm(1)] \item If $q$ is an algebraic integer of full rank and $n$ is a fixed positive integer, then there are only finitely many $m$ such that $\disc\left(\BbZ[q^m]\right)=\disc\left(\BbZ[q^n]\right)$. \item If $q$ and $r$ are algebraic integers of degree $d$ of full rank and $\BbZ[q^n] = \BbZ[r^n]$ for infinitely many $n$, then either $q = \omega r'$ or $q={\rm Norm}(r)^{2/d}\omega/r'$, where $r'$ is some conjugate of $r$ and $\omega$ is some root of unity. \item Let $r$ be an algebraic integer of degree at most $3$. Then there are at most $40$ Pisot numbers $q$ such that $\BbZ[q] = \BbZ[r]$. \item There are only finitely many Pisot-cyclotomic numbers of any fixed order. \end{compactenum}

Keywords:algebraic integers, Pisot numbers, full rank, discriminant
Categories:11R04, 11R06

107. CJM 2009 (vol 61 pp. 465)

Woodford, Roger
On Partitions into Powers of Primes and Their Difference Functions
In this paper, we extend the approach first outlined by Hardy and Ramanujan for calculating the asymptotic formulae for the number of partitions into $r$-th powers of primes, $p_{\mathbb{P}^{(r)}}(n)$, to include their difference functions. In doing so, we rectify an oversight of said authors, namely that the first difference function is perforce positive for all values of $n$, and include the magnitude of the error term.

Categories:05A17, 11P81

108. CJM 2009 (vol 61 pp. 165)

Laurent, Michel
Exponents of Diophantine Approximation in Dimension Two
Let $\Theta=(\alpha,\beta)$ be a point in $\bR^2$, with $1,\alpha, \beta$ linearly independent over $\bQ$. We attach to $\Theta$ a quadruple $\Omega(\Theta)$ of exponents that measure the quality of approximation to $\Theta$ both by rational points and by rational lines. The two ``uniform'' components of $\Omega(\Theta)$ are related by an equation due to Jarn\'\i k, and the four exponents satisfy two inequalities that refine Khintchine's transference principle. Conversely, we show that for any quadruple $\Omega$ fulfilling these necessary conditions, there exists a point $\Theta\in \bR^2$ for which $\Omega(\Theta) =\Omega$.

Categories:11J13, 11J70

109. CJM 2009 (vol 61 pp. 3)

Behrend, Kai; Dhillon, Ajneet
Connected Components of Moduli Stacks of Torsors via Tamagawa Numbers
Let $X$ be a smooth projective geometrically connected curve over a finite field with function field $K$. Let $\G$ be a connected semisimple group scheme over $X$. Under certain hypotheses we prove the equality of two numbers associated with $\G$. The first is an arithmetic invariant, its Tamagawa number. The second is a geometric invariant, the number of connected components of the moduli stack of $\G$-torsors on $X$. Our results are most useful for studying connected components as much is known about Tamagawa numbers.

Categories:11E, 11R, 14D, 14H

110. CJM 2009 (vol 61 pp. 141)

Green, Ben; Konyagin, Sergei
On the Littlewood Problem Modulo a Prime
Let $p$ be a prime, and let $f \from \mathbb{Z}/p\mathbb{Z} \rightarrow \mathbb{R}$ be a function with $\E f = 0$ and $\Vert \widehat{f} \Vert_1 \leq 1$. Then $\min_{x \in \Zp} |f(x)| = O(\log p)^{-1/3 + \epsilon}$. One should think of $f$ as being ``approximately continuous''; our result is then an ``approximate intermediate value theorem''. As an immediate consequence we show that if $A \subseteq \Zp$ is a set of cardinality $\lfloor p/2\rfloor$, then $\sum_r |\widehat{1_A}(r)| \gg (\log p)^{1/3 - \epsilon}$. This gives a result on a ``mod $p$'' analogue of Littlewood's well-known problem concerning the smallest possible $L^1$-norm of the Fourier transform of a set of $n$ integers. Another application is to answer a question of Gowers. If $A \subseteq \Zp$ is a set of size $\lfloor p/2 \rfloor$, then there is some $x \in \Zp$ such that \[ | |A \cap (A + x)| - p/4 | = o(p).\]

Categories:42A99, 11B99

111. CJM 2008 (vol 60 pp. 1306)

Mui\'c, Goran
Theta Lifts of Tempered Representations for Dual Pairs $(\Sp_{2n}, O(V))$
This paper is the continuation of our previous work on the explicit determination of the structure of theta lifts for dual pairs $(\Sp_{2n}, O(V))$ over a non-archimedean field $F$ of characteristic different than $2$, where $n$ is the split rank of $\Sp_{2n}$ and the dimension of the space $V$ (over $F$) is even. We determine the structure of theta lifts of tempered representations in terms of theta lifts of representations in discrete series.

Categories:22E35, 22E50, 11F70

112. CJM 2008 (vol 60 pp. 1406)

Ricotta, Guillaume; Vidick, Thomas
Hauteur asymptotique des points de Heegner
Geometric intuition suggests that the N\'{e}ron--Tate height of Heegner points on a rational elliptic curve $E$ should be asymptotically governed by the degree of its modular parametrisation. In this paper, we show that this geometric intuition asymptotically holds on average over a subset of discriminants. We also study the asymptotic behaviour of traces of Heegner points on average over a subset of discriminants and find a difference according to the rank of the elliptic curve. By the Gross--Zagier formulae, such heights are related to the special value at the critical point for either the derivative of the Rankin--Selberg convolution of $E$ with a certain weight one theta series attached to the principal ideal class of an imaginary quadratic field or the twisted $L$-function of $E$ by a quadratic Dirichlet character. Asymptotic formulae for the first moments associated with these $L$-series and $L$-functions are proved, and experimental results are discussed. The appendix contains some conjectural applications of our results to the problem of the discretisation of odd quadratic twists of elliptic curves.

Categories:11G50, 11M41

113. CJM 2008 (vol 60 pp. 1267)

Blake, Ian F.; Murty, V. Kumar; Xu, Guangwu
Nonadjacent Radix-$\tau$ Expansions of Integers in Euclidean Imaginary Quadratic Number Fields
In his seminal papers, Koblitz proposed curves for cryptographic use. For fast operations on these curves, these papers also initiated a study of the radix-$\tau$ expansion of integers in the number fields $\Q(\sqrt{-3})$ and $\Q(\sqrt{-7})$. The (window) nonadjacent form of $\tau$-expansion of integers in $\Q(\sqrt{-7})$ was first investigated by Solinas. For integers in $\Q(\sqrt{-3})$, the nonadjacent form and the window nonadjacent form of the $\tau$-expansion were studied. These are used for efficient point multiplications on Koblitz curves. In this paper, we complete the picture by producing the (window) nonadjacent radix-$\tau$ expansions for integers in all Euclidean imaginary quadratic number fields.

Keywords:algebraic integer, radix expression, window nonadjacent expansion, algorithm, point multiplication of elliptic curves, cryptography
Categories:11A63, 11R04, 11Y16, 11Y40, 14G50

114. CJM 2008 (vol 60 pp. 975)

Boca, Florin P.
An AF Algebra Associated with the Farey Tessellation
We associate with the Farey tessellation of the upper half-plane an AF algebra $\AA$ encoding the ``cutting sequences'' that define vertical geodesics. The Effros--Shen AF algebras arise as quotients of $\AA$. Using the path algebra model for AF algebras we construct, for each $\tau \in \big(0,\frac{1}{4}\big]$, projections $(E_n)$ in $\AA$ such that $E_n E_{n\pm 1}E_n \leq \tau E_n$.

Categories:46L05, 11A55, 11B57, 46L55, 37E05, 82B20

115. CJM 2008 (vol 60 pp. 1028)

Hamblen, Spencer
Lifting $n$-Dimensional Galois Representations
We investigate the problem of deforming $n$-dimensional mod $p$ Galois representations to characteristic zero. The existence of 2-dimensional deformations has been proven under certain conditions by allowing ramification at additional primes in order to annihilate a dual Selmer group. We use the same general methods to prove the existence of $n$-dimensional deformations. We then examine under which conditions we may place restrictions on the shape of our deformations at $p$, with the goal of showing that under the correct conditions, the deformations may have locally geometric shape. We also use the existence of these deformations to prove the existence as Galois groups over $\Q$ of certain infinite subgroups of $p$-adic general linear groups.


116. CJM 2008 (vol 60 pp. 1149)

Petersen, Kathleen L.; Sinclair, Christopher D.
Conjugate Reciprocal Polynomials with All Roots on the Unit Circle
We study the geometry, topology and Lebesgue measure of the set of monic conjugate reciprocal polynomials of fixed degree with all roots on the unit circle. The set of such polynomials of degree $N$ is naturally associated to a subset of $\R^{N-1}$. We calculate the volume of this set, prove the set is homeomorphic to the $N-1$ ball and that its isometry group is isomorphic to the dihedral group of order $2N$.

Categories:11C08, 28A75, 15A52, 54H10, 58D19

117. CJM 2008 (vol 60 pp. 734)

Baba, Srinath; Granath, H\aa kan
Genus 2 Curves with Quaternionic Multiplication
We explicitly construct the canonical rational models of Shimura curves, both analytically in terms of modular forms and algebraically in terms of coefficients of genus 2 curves, in the cases of quaternion algebras of discriminant 6 and 10. This emulates the classical construction in the elliptic curve case. We also give families of genus 2 QM curves, whose Jacobians are the corresponding abelian surfaces on the Shimura curve, and with coefficients that are modular forms of weight 12. We apply these results to show that our $j$-functions are supported exactly at those primes where the genus 2 curve does not admit potentially good reduction, and construct fields where this potentially good reduction is attained. Finally, using $j$, we construct the fields of moduli and definition for some moduli problems associated to the Atkin--Lehner group actions.

Keywords:Shimura curve, canonical model, quaternionic multiplication, modular form, field of moduli
Categories:11G18, 14G35

118. CJM 2008 (vol 60 pp. 790)

Blasco, Laure
Types, paquets et changement de base : l'exemple de $U(2,1)(F_0)$. I. Types simples maximaux et paquets singletons
Soit $F_0$ un corps local non archim\'edien de caract\'eristique nulle et de ca\-rac\-t\'eristique r\'esiduelle impaire. J. Rogawski a montr\'e l'existence du changement de base entre le groupe unitaire en trois variables $U(2,1)(F_{0})$, d\'efini relativement \`a une extension quadratique $F$ de $F_{0}$, et le groupe lin\'eaire $GL(3,F)$. Par ailleurs, nous avons d\'ecrit les repr\'esentations supercuspidales irr\'eductibles de $U(2,1)(F_{0})$ comme induites \`a partir d'un sous-groupe compact ouvert de $U(2,1)(F_{0})$, description analogue \`a celle des repr\'esentations admissibles irr\'eductibles de $GL(3,F)$ obtenue par C. Bushnell et P. Kutzko. A partir de ces descriptions, nous construisons explicitement le changement de base des repr\'esentations tr\`es cuspidales de $U(2,1)(F_{0})$.

Categories:22E50, 11F70

119. CJM 2008 (vol 60 pp. 532)

Clark, Pete L.; Xarles, Xavier
Local Bounds for Torsion Points on Abelian Varieties
We say that an abelian variety over a $p$-adic field $K$ has anisotropic reduction (AR) if the special fiber of its N\'eron minimal model does not contain a nontrivial split torus. This includes all abelian varieties with potentially good reduction and, in particular, those with complex or quaternionic multiplication. We give a bound for the size of the $K$-rational torsion subgroup of a $g$-dimensional AR variety depending only on $g$ and the numerical invariants of $K$ (the absolute ramification index and the cardinality of the residue field). Applying these bounds to abelian varieties over a number field with everywhere locally anisotropic reduction, we get bounds which, as a function of $g$, are close to optimal. In particular, we determine the possible cardinalities of the torsion subgroup of an AR abelian surface over the rational numbers, up to a set of 11 values which are not known to occur. The largest such value is 72.

Categories:11G10, 14K15

120. CJM 2008 (vol 60 pp. 491)

Bugeaud, Yann; Mignotte, Maurice; Siksek, Samir
A Multi-Frey Approach to Some Multi-Parameter Families of Diophantine Equations
We solve several multi-parameter families of binomial Thue equations of arbitrary degree; for example, we solve the equation \[ 5^u x^n-2^r 3^s y^n= \pm 1, \] in non-zero integers $x$, $y$ and positive integers $u$, $r$, $s$ and $n \geq 3$. Our approach uses several Frey curves simultaneously, Galois representations and level-lowering, new lower bounds for linear forms in $3$ logarithms due to Mignotte and a famous theorem of Bennett on binomial Thue equations.

Keywords:Diophantine equations, Frey curves, level-lowering, linear forms in logarithms, Thue equation
Categories:11F80, 11D61, 11D59, 11J86, 11Y50

121. CJM 2008 (vol 60 pp. 481)

Breuer, Florian; Im, Bo-Hae
Heegner Points and the Rank of Elliptic Curves over Large Extensions of Global Fields
Let $k$ be a global field, $\overline{k}$ a separable closure of $k$, and $G_k$ the absolute Galois group $\Gal(\overline{k}/k)$ of $\overline{k}$ over $k$. For every $\sigma\in G_k$, let $\ks$ be the fixed subfield of $\overline{k}$ under $\sigma$. Let $E/k$ be an elliptic curve over $k$. It is known that the Mordell--Weil group $E(\ks)$ has infinite rank. We present a new proof of this fact in the following two cases. First, when $k$ is a global function field of odd characteristic and $E$ is parametrized by a Drinfeld modular curve, and secondly when $k$ is a totally real number field and $E/k$ is parametrized by a Shimura curve. In both cases our approach uses the non-triviality of a sequence of Heegner points on $E$ defined over ring class fields.


122. CJM 2008 (vol 60 pp. 412)

Nguyen-Chu, G.-V.
Quelques calculs de traces compactes et leurs transform{ées de Satake
On calcule les restrictions {\`a} l'alg{\`e}bre de Hecke sph{\'e}rique des traces tordues compactes d'un ensemble de repr{\'e}sentations explicitement construites du groupe $\GL(N, F)$, o{\`u} $F$ est un corps $p$-adique. Ces calculs r\'esolve en particulier une question pos{\'e}e dans un article pr\'ec\'edent du m\^eme auteur.

Categories:22E50, 11F70

123. CJM 2008 (vol 60 pp. 208)

Ramakrishna, Ravi
Constructing Galois Representations with Very Large Image
Starting with a 2-dimensional mod $p$ Galois representation, we construct a deformation to a power series ring in infinitely many variables over the $p$-adics. The image of this representation is full in the sense that it contains $\SL_2$ of this power series ring. Furthermore, all ${\mathbb Z}_p$ specializations of this deformation are potentially semistable at $p$.

Keywords:Galois representation, deformation

124. CJM 2007 (vol 59 pp. 1121)

Alayont, Feryâl
Meromorphic Continuation of Spherical Cuspidal Data Eisenstein Series
Meromorphic continuation of the Eisenstein series induced from spherical, cuspidal data on parabolic subgroups is achieved via reworking Bernstein's adaptation of Selberg's third proof of meromorphic continuation.

Categories:11F72, 32N10, 32D15

125. CJM 2007 (vol 59 pp. 1284)

Fukshansky, Lenny
On Effective Witt Decomposition and the Cartan--Dieudonn{é Theorem
Let $K$ be a number field, and let $F$ be a symmetric bilinear form in $2N$ variables over $K$. Let $Z$ be a subspace of $K^N$. A classical theorem of Witt states that the bilinear space $(Z,F)$ can be decomposed into an orthogonal sum of hyperbolic planes and singular and anisotropic components. We prove the existence of such a decomposition of small height, where all bounds on height are explicit in terms of heights of $F$ and $Z$. We also prove a special version of Siegel's lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces. Finally, we prove an effective version of the Cartan--Dieudonn{\'e} theorem. Namely, we show that every isometry $\sigma$ of a regular bilinear space $(Z,F)$ can be represented as a product of reflections of bounded heights with an explicit bound on heights in terms of heights of $F$, $Z$, and $\sigma$.

Keywords:quadratic form, heights
Categories:11E12, 15A63, 11G50
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